Multi-asset Black–Scholes model as a variable second class constrained dynamical system

In this paper, we study the multi-asset Black–Scholes model from a structural point of view. For this, we interpret the multi-asset Black–Scholes equation as a multidimensional Schrödinger one particle equation. The analysis of the classical Hamiltonian and Lagrangian mechanics associated with this quantum model implies that, in this system, the canonical momentums cannot always be written in terms of the velocities. This feature is a typical characteristic of the constrained system that appears in the high-energy physics. To study this model in the proper form, one must apply Dirac’s method for constrained systems. The results of the Dirac’s analysis indicate that in the correlation parameters space of the multi-assets model, there exists a surface (called the Kummer surface ΣK, where the determinant of the correlation matrix is null) on which the constraint number can vary. We study in detail the cases with N=2 and N=3 assets. For these cases, we calculate the propagator of the multi-asset Black–Scholes equation and show that inside the Kummer ΣK surface the propagator is well defined, but outside ΣK the propagator diverges and the option price is not well defined. On ΣK the propagator is obtained as a constrained path integral and their form depends on which region of the Kummer surface the correlation parameters lie. Thus, the multi-asset Black–Scholes model is an example of a variable constrained dynamical system, and it is a new and beautiful property that had not been previously observed.